Within the spectral thermodynamic limit approach, the spectral DOS \(f(\eta )\)  of an equilibrium (uniform) soliton gas is defined via the nonlinear dispersion relations. The nonlinear dispersion relations for soliton gas are
derived by applying the thermodynamic spectral limit to the wavenumber-frequency relations for the finite-gap KdV solutions [1]} (see [2]}, [3]} for details). Let \(\eta =\sqrt{-\lambda }\) , then the spectral DOS \(f(\eta )\)  and the corresponding spectral flux density \(v(\eta )\)  satisfy
\(\int _\Gamma \log \left|\frac{\eta + \mu }{\eta -\mu }\right| f(\mu )\mathrm {d}\mu + \sigma (\eta ) f(\eta )= \eta \, ,\) 
\(\int _\Gamma \log \left|\frac{\eta + \mu }{\eta -\mu }\right|v(\mu ) \mathrm {d}\mu + \sigma (\eta )v(\eta )=4\eta ^3 \, .\) 
